The charges sought are under Section 3 (1)(c) of the act, which "covers entering prohibited places, taking images or handling secret official documents that 'might be or is intended to be, directly or indirectly, useful to an enemy,'" the news agency said.
Reuters Editor-in-Chief Stephen Adler said the agency was "extremely disappointed" at what he termed an "wholly unwarranted, blatant attack on press freedom."
The reporters previously appeared in court in December, but no charges were sought. Following that appearance, they were permitted to meet relatives and a lawyer for the first time since their arrest, the news agency reported. The case has been adjourned until January 23.
Govt: Reporters 'illegally acquired information'
In a statement to Reuters in December, the Myanmar government said the reporters had been detained after they "illegally acquired information with the intention to share it with foreign media."
An image of the two men in handcuffs with their faces partially obscured was published on the Ministry of Information's page.
According to the US Embassy in Myanmar at the time of their detention, the journalists were arrested "after they were invited to meet with police officials."
The two reporters were working in Rakhine State, the location of alleged ethnic cleansing of the Rohingya, the state's minority Muslim population. The Myanmar government says the army has been conducting operations against militants who attacked border posts.
Since the start of the military operation in August last year, more than 650,000 Rohingya had fled across the border to neighboring Bangladesh.
The aid agency's death toll far surpasses estimates from Myanmar's government, which has put the figure in the hundreds.
On Tuesday, former US President Bill Clinton added his voice to calls for the men to be freed.
"A free press is critical to a free society - the detention of journalists anywhere is unacceptable. The Reuters journalists being held in Myanmar should be released immediately," Clinton said in a tweet.